If you were brought up on pages of ‘hard sums’ you may think maths is difficult
and boring. Worse than that you may think you’re not very good at it!!! That’s a
real shame because it is a fun subject and for most people, if they are taught to
understand numbers, they should be reasonably easy to grasp and use.
If what you remember as maths is pages of sums you may sometimes feel
confused when your child’s maths book contains writing, pictures, diagrams,
jottings or blank number lines and not many ‘formal calculations’. Certainly
younger children, up to year 3, will record calculations in a variety of ways that
do not necessarily look like the kind of ‘sums’ you remember. This is because
written calculations are not the ultimate aim: the aim is for children to do
calculations in their heads and, if the numbers are too large, to use a way of
writing them down that helps their thinking.
As children develop their knowledge and understanding through years 3,4,5 and
6 teachers will be asking them to look at any calculation and ask “Can I do this in
my head?” Sometimes this will need to be supported by a drawing, diagram or
numerical jotting (notes). If they can’t do it largely in their heads they should
be looking for the most suitable written method or, during years 5 and 6, using a
calculator for more complex calculations.
Here we try, as simply as possible, to help you to help your children. We take
you through the ideas relating to children’s number development from the
earliest counting and mental skills to their recording of calculations to support
thinking. If you’ve never felt very confident with numbers and calculations this
also might help you, you never know! Read on and see.
September 2005
Leicestershire Numeracy Team
When children are in years 1 and 2 they are not expected to do vertical sums
like
but that doesn’t mean they won’t learn that 6+4=10.
6
+4
10
They will be doing a daily mixture of practical, mental and oral work including
lots of counting, talking about numbers and using numbers in real life activities.
They will begin to record what they’ve done with pictures and numbers. These
recordings will help them to understand what is happening and to show how
they’ve worked something out. Here are two examples of early recording.
This next example shows how different children have worked out and recorded
the answer to the same problem about the children in the class.
These diagrams and jottings help the children to see what is happening to the
numbers and to use some facts they already know to help them work out others.
In years 3 and 4 children will carry on using horizontal recording of addition and
subtraction to support their mental calculations. The example below shows two
ways of adding 76 and 93. The first splits the numbers into tens and ones
(units) then adds the tens followed by the ones to give 169. The second example
uses the idea of rounding to 100 by taking 7 from 76 and adding it to the 93 to
make it 100 so making the addition easier.
September 2005
Leicestershire Numeracy Team
Children will also continue to use drawings, diagrams and blank number lines to
support their thinking, as below.
Towards the end of year 3 and into year 4 most children will be taught written
methods, including vertical addition, using an ‘expanded method’, and subtraction
using number lines by counting on (complementary addition). This form of
recording will be used for those calculations that they can’t do ‘in their heads’.
‘Expanded methods’ are ways of recording that make the process of adding the
different digits clear to children. These methods build on the mental methods
they have been learning and should help children to understand what is
happening. Here is an example of adding using an expanded method.
The blue team’s score of 287 points is
increased by 145 points.What’s the new score?
287
+ 145
432
200
100
300
+ 80
+ 40
+120
+7
+5
+12
This is the same question set out in a different way but using the same ideas. It
includes an explanation of what might be said to describe the adding.
The blue team’s score of 287 points
is increased by 145 points. What is
the new score?
+
September 2005
2 8 7
1 4 5
1 2
1 2 0
3 0 0
4 3 2
Explanation……
The language used is very important to help children
understand the size of numbers being added
(e.g. is it seven or seventy or seven hundred)
seven plus five equals twelve
eighty plus forty equals one hundred and twenty
two hundred plus one hundred equals three hundred
finally three hundred plus one hundred and twenty
plus twelve equals four hundred and thirty two
Leicestershire Numeracy Team
These methods mean that children may have to write a little more at this stage
but, because it helps and supports their understanding, it enables them to
become much more confident and quicker in the long run.
Here is an example of a subtraction problem involving 3 digit numbers:
754 – 286 = 468
+400
+14
286
300
+54
754
700
OR
754 - 286 = 468
14 (300)
400 (700)
54 (754)
468
can be refined to
14 (300)
454 (754)
468
Here you can see that children are using an image
of a number line to support their thinking to work
out this subtraction. Children are encouraged to
find the difference between 286 and 754 by
counting on, just as shopkeepers work out change,
and how you might calculate mentally. Beginning at
286, 14 is needed to count on to the next ‘friendly’
number, 300; then a further 400 to land on 700;
and finally another 54. All of the jumps total 468.
Without the number line the calculation might look
like the second example. This method of
subtraction is known as ‘complementary addition’.
Try to decide how you would do the following calculations. Would you do them in
your head, write them down or use a calculator? The notes may change your
mind but don’t read them until you’ve had a go at the calculation.
The total of all
45+99
3006-2999
2.3 + 6.99
4532-3768
the numbers
from 1 to 10
This is easy to do if
you think of 99 being
one less than 100. So
add 100 to make 145
and then take away 1
to give 144
These numbers are
very close on the
number line. We need
1 to get from 2999
to 3000 and 6 more
to get to 3006 so the
difference is 7. So
much easier than
doing a vertical sum!
This addition
involving decimals
may look hard but it’s
easier if you think of
it as money.
£2.30 add £6.99.
So add £7 to get
£9.30 and take away
1p to get £9.29
Easy!
This, for most
people, needs a pencil
and paper or
calculator if speed is
important.
This is easy if you think
of pairs of numbers
making 10
1 + 9 = 10
2 + 8 = 10
3 + 7 = 10
4 + 6 = 10
Finally
5 +10 =15
giving a total of 55
It is important that calculations are presented in problems or horizontally, as
above, to encourage children to think about the numbers as a whole, what they
mean, what a sensible answer might be and the best method of working them
out. It’s all about giving children confidence with and control over numbers.
September 2005
Leicestershire Numeracy Team
Did anyone ever tell you that you only needed to learn about half of the
multiplication tables in order to know them all? If they didn’t it was a bit mean
because if you know 3x4=12 you also know 4x3=12, so why learn it twice?
Did anyone ever say that once you knew the 2 times table, which is only double
the 1 times, then the 4 times was easy because you just double the 2 times?
Then you can double the 4 times to get the 8 times. The 3 times doubles to the
6 times and 12 times, the ten times can be halved to give 5 times and so on. This
uses the knowledge children are developing through addition and subtraction
and makes important connections for them. This chart shows how this works for
the 2x, 4x and 8x tables.
x
2times
4times
8times
1
2
4
8
2
4
8
16
3
6
12
24
4
8
16
32
5
10
20
40
6
12
24
48
7
14
28
56
8
16
32
64
9
18
36
72
10
20
40
80
double
It’s also possible that you weren’t told that you knew your division tables. If you
were shown that division was the opposite of multiplication you will understand
that knowing 3x4=12 or 4x3=12 also means you know 124=3 and 123=4. So
knowing one number fact, like 3x4=12, immediately means we know at least four.
But did you also realise that knowing any one of these facts helps you to know a
lot more than four without actually learning them? Read on.
The early work children do will introduce them to the ideas of multiplication and
division. They will be counting in different patterns, helped to see how
multiplication is repeated addition and division is repeated subtraction, shown
how division is the opposite of multiplication and taught to understand place
value (that in 234 the 2 is 200, the 3 is 30 and the 4 is 4 ones (units)). This
knowledge and understanding, with much of the work being done in their heads,
opens up a whole world of facts for them and they don’t all have to be
memorised. That can make dealing with numbers feel a lot easier.
September 2005
Leicestershire Numeracy Team
The following chart shows something of what this means.
If you know 4x5= 20 what else do you know?
5x4=20
204=5
205=4
4+4+4+4+4=20
5+5+5+5=20
and using knowledge of place value
5x40=200
4x50=200
20050=4
20040=5
40x50= 2000 and so on
combined with knowing the multiples of ten, which
are easy, we can work out that
5x44=220 (5x40=200 and 5x4=20…200+20=220)
therefore
2205=44
and so it goes on. Try it for yourself..in your head!
The ability to do what you’ve just seen, developed gradually through years 1,2,3
& 4 helps children in years 4,5 & 6 to move on with confidence to multiplication
and division of bigger numbers including those involving decimals and as with
addition and subtraction, the questions will usually be presented to children as
word problems or, horizontally, as calculations. The children should then be
encouraged to work them out mentally if they can (supported by drawings,
diagrams, number jottings if necessary) or, if they can’t, to use the most
suitable written method they know.
In years 1 & 2 the children will be recording to demonstrate how they have done
something and to show that they’ve understood what is happening, as below.
2x3 cats =6 cats or
3x2 cats =6 cats
September 2005
2 lots of 3
apples makes 6
apples.
Leicestershire Numeracy Team
In years 3 & 4 the children will begin to use expanded methods to help them
deal with calculations that they can’t do in their heads. At this stage it will
mostly involve multiplying and dividing 2 digit numbers by a single digit (72x6 or
854). When dividing they will learn about and use remainders.
The expanded method for multiplying is often called the grid method. It uses
the mental skills and the knowledge children have been learning and will help
most children to move, with understanding, to the ‘compact’ method you may
know.
This chart shows ‘the grid method’. You can see, as with addition and
subtraction expanded methods, it uses knowledge of number facts and the idea
of splitting a number into its parts (place value) to help understanding of the
process.
How many sweets do I need for 24 party
bags if each is to have 6 sweets?
x
6
20
120
4
24
= 144 sweets
You will see the 24 has been split into 20 and
4, each has been multiplied by 6 mentally and
the two numbers added to give the final total.
Many children will eventually develop the
ability to do this kind of calculation totally in
their heads.
Here is a slightly more difficult example.
How much does it cost if I buy 9 books at
72p each?
x
9
70
630
2
18
= 648
It costs 648p or £6.48
The expanded method for division is often called ‘chunking’ and really just
involves partitioning the number into helpful ‘chunks’ related to the number you
are dividing by (divisor) or counting on/taking away chunks of the same size until
you run out. It uses the fact that division is repeated subtraction of the same
size group. So 204=5 involves subtracting 4s from 20 until it’s been used up or
counting on in fours until you reach 20. You can do this 5 times.
September 2005
Leicestershire Numeracy Team
Here is a more difficult example showing how larger chunks are used to speed
up the process. Again this method uses and builds on the ideas explained earlier.
72 pears are packed in boxes of 6. How many boxes would
there be?
First partition the 72 into chunks related to multiples of
the divisor
72 = 60 + 12
then divide each part by 6
60 ÷ 6 = 10
12 ÷ 6 = 2
then add the 10 and the 2 to get 12
So 12 boxes of pears could be packed
As they move into years 5 & 6 children will still be encouraged to choose the
most suitable method of calculation, mentally if possible. Where this is not
10 + 2 = 12
possible they will be using expanded or compact methods and a calculator for
more complex and involved work.
Most children will be expected to multiply 3 digits x 1 digit and 3 digits x 2
digits using a written method. Here are some examples.
How many hours are there in the year 2003?
This means we have to do 365x24…a calculation
you may find quite hard. Here it is expanded and
using lots of mental skills…but none of them
difficult.
x
20
4
300
60
5
6000 1200 100
1200 240 20
= 7300
= 1460
So 20x365= 7300
4x365= 1460
giving a total of 8760 hours with no
difficult calculation to do.
More complex division will involve dividing 3 digit numbers by a 1 digit number
and 3 digit numbers by a 2 digit number. With division, as with all calculation,
it’s important to think about what the actual problem is asking when you come to
give an answer. This is shown up in the second example of expanded division or
chunking involving buses for a school trip.
September 2005
Leicestershire Numeracy Team
458 ÷ 3
partition 458 into multiples of the
divisor
458 = 300 + 150 + 8
458 stickers are shared between 3 children. How
many does each get?
4
3
1
1
300 ÷ 3 = 100
150 ÷ 3 = 50
8 ÷ 3 = 2r2
add these together
100 + 50 + 2r2 = 152 remainder 2
5
0
5
5
8
0
8
0
8
6
2
that’s 100 x3
left
that’s 50 x3
left
that’s
2 x3
left over
So each gets 100+50+2 or 152 stickers with 2
left over.
458  3 = 152 remainder 2
432 children and adults are going on a school trip. If
each bus takes 30 people how many are needed?
4
3
1
1
3
0
3
2
1
2
0
2
0
2
people going
that’s 10 x30
left
that’s 4 x30
left
or10 buses
or 4 buses
1 bus
So we see that the calculation would result in
432  30 = 14 remainder 12
This is not a good answer for this question because the
12 people left over would need another bus or they
couldn’t go!
So we see that 15 buses are needed…or some cars.
When the children really understand these expanded methods they can be
shown how they are developed into a compact method. Remember though that
the expanded methods are perfectly good ways of working out an answer if the
children feel more comfortable and therefore find it easier. They give the same
answer and it can often be quicker if they are confident about what they are
doing.
These methods are very useful when children are extending their work to
numbers involving decimals but we’ll leave that for another day!
September 2005
Leicestershire Numeracy Team
Hopefully reading this has made you feel a little more confident and
comfortable with numbers and calculations yourself, if you weren’t already, and
therefore better able to help your child. Remember if you have any concerns or
questions it’s always better to talk to your child’s teacher rather than pulling
the children in different directions or worrying about what’s going on. Children
(and adults) need to feel confident with numbers and to enjoy playing with them
and using them, that’s really what it’s all about. It then means using them for
everyday purposes becomes a doddle rather than a threat.
Finally, have a go at these using expanded methods. Trying things out helps
understanding and the answers are at the bottom of the box for you so you’ll
know if you are correct. One of them at least could be done completely in your
head…maybe! Remember we want children to decide on the best method,
mentally if possible, or the most suitable, manageable written method if not.
Have fun.
calculation
a) 27 x 7
b)
238 x 45
working
calculation
c)
176  6
d)
346 16
working
answers
a)189
Several images in this document are used with
kind permission of QCA.
September 2005
c)29 rem 2
b)10710
d)21 rem 10
Numeracy in Leicestershire 2005
Leicestershire Numeracy Team